First-order error propagation formulae for the Gassmann relations

The Gassmann relations of poroelasticity provide a connection between the dry and the saturated elastic moduli of porous rock and are useful in a variety of petroleum geoscience applications. Because some uncertainty is usually associated with the input parameters, the propagation of error in the inputs into the final moduli estimates is immediately of interest. Two common approaches to error propagation include: a first-order Taylor series expansion and Monte-Carlo methods. The Taylor series approach requires derivatives, which are obtained either analytically or numerically and is usually limited to a first-order analysis. The formulae for analytical derivatives were often prohibitively complicated before modern symbolic computation packages became prevalent but they are now more accessible. We apply this method and present formulae for uncertainty in the predicted bulk and shear moduli for two forms of the Gassmann relations. Numerical results obtained with these uncertainty formulae are compared with Monte-Carlo calculations as a form of validation and to illustrate the relative characteristics of the two approaches. Particular emphasis is given to the problem of correlated variables, which are often ignored in naïve approaches to error analysis. Going out to the error level that the two methods were compared, the means agree and the variance of the Monte Carlo method for bulk modulus grows with input error.